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Two efficient methods for solving Schlömilch’s integral equation
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Purpose

In this paper, the exact solutions of the Schlömilch’s integral equation and its linear and non-linear generalized formulas with application are solved by using two efficient iterative methods. The Schlömilch’s integral equations have many applications in atmospheric, terrestrial physics and ionospheric problems. They describe the density profile of electrons from the ionospheric for awry occurrence of the quasi-transverse approximations. The paper aims to discuss these issues.

Design/methodology/approach

First, the authors apply a regularization method combined with the standard homotopy analysis method to find the exact solutions for all forms of the Schlömilch’s integral equation. Second, the authors implement the regularization method with the variational iteration method for the same purpose. The effectiveness of the regularization-Homotopy method and the regularization-variational method is shown by using them for several illustrative examples, which have been solved by other authors using the so-called regularization-Adomian method.

Findings

The implementation of the two methods demonstrates the usefulness in finding exact solutions.

Practical implications

The authors have applied the developed methodology to the solution of the Rayleigh equation, which is an important equation in fluid dynamics and has a variety of applications in different fields of science and engineering. These include the analysis of batch distillation in chemistry, scattering of electromagnetic waves in physics, isotopic data in contaminant hydrogeology and others.

Originality/value

In this paper, two reliable methods have been implemented to solve several examples, where those examples represent the main types of the Schlömilch’s integral models. Each method has been accompanied with the use of the regularization method. This process constructs an efficient dealing to get the exact solutions of the linear and non-linear Schlömilch’s integral equation which is easy to implement. In addition to that, the accompanied regularization method with each of the two used methods proved its efficiency in handling many problems especially ill-posed problems, such as the Fredholm integral equation of the first kind.

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Publication Date
Mon Aug 17 2020
Journal Name
International Journal Of Applied Mechanics And Engineering
Analysis of Structural Concrete Bar Members Based on Secant Stiffness Methods
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Abstract<p>In this paper, the behavior of structural concrete linear bar members was studied using numerical model implemented in a computer program written in MATLAB. The numerical model is based on the modified version of the procedure developed by Oukaili. The model is based on real stress-strain diagrams of concrete and steel and their secant modulus of elasticity at different loading stages. The behavior presented by normal force-axial strain and bending moment-curvature relationships is studied by calculating the secant sectional stiffness of the member. Based on secant methods, this methodology can be easily implemented using an iterative procedure to solve non-linear equations. A compari</p> ... Show More
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Publication Date
Wed Feb 29 2012
Journal Name
Al-khwarizmi Engineering Journal
Effect of Laser Cutting Methods on Hole Deviation and Surface Integrity
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In the present work usedNd:YAG laser systems of different output characteristic were employed to study the drilling process of material used in scientific and industrial fields. This material include Manganese hard steel. Our study went into the affecting parameters in drilling of Manganese hard steel by laser. Drilling process is achieved through material absorption of part of the incident laser beam. It is the resultant of interfering both, laser beam and material properties and the focusing conditions of the beam. The results as shown that the increase in the laser pulse energy over the used level has raised the hole diameter, depth and increased the hole taper. In addition to that a hole taper was affected by the laser energy, the fo

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Publication Date
Thu Oct 01 2015
Journal Name
Journal Of Educational And Psychological Researches
Recent trends of Arabic language teaching methods in contemporary education environment
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Arabic language the timeless language of the Qur'an ((I made it read Arabic you may understand)) (Al-Zukhruf No 3). It is a way to express his thoughts and feelings and capable of giving constant absorbed the needs of the times and meet the General requirements and interactive needs of life to achieve social and intellectual aspects.    

The aim of current research to identify what teaching the contemporary trends in teaching, what the nature of knowledge and teaching methods, how do you rate the modern Arabic language teaching methods, the criteria for evaluating the effectiveness of teaching where the researcher has reached some recommendations: 1. establish the basics of environmental ed

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Publication Date
Tue Dec 01 2009
Journal Name
Bulletin Of The Iraq Natural History Museum (p-issn: 1017-8678 , E-issn: 2311-9799)
RECORD OF TWO SPECIES OF THE MONOGENETIC TREMATODES GENUS DACTYLOGYRUS FOR THE FIRST TIME IN IRAQ ON GILLS OF THE CYPRINID FISH ALBURNUS CAERULEUS
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For the first time in Iraq, two species of monogenetic tretamtodes of the genus Dactylogyrus were recorded from gills of Alburnus caeruleus from Tigris river at Al-Zaafaraniya, south of Baghdad during June 1995. The first species, Dactylogyrus sphyrna is characterized by having the seventh pair of marginal hooklets almost twice as large as other marginal hooklets, powerful inner and outer processes of median hooks especially the inner one which is expanded terminally, one connecting bar and long spirally twisted copulatory organ. The second species, Daclytogyrus phoxini differs from the first one by having marginal hooklets of the same size, inner and outer processes of median hooks are not powerful, two connecting bars

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Publication Date
Wed Aug 01 2018
Journal Name
Collegian
Behaviour change interventions to improve medication adherence in patients with cardiac disease: Protocol for a mixed methods study including a pilot randomised controlled trial
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Crossref (6)
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Publication Date
Sun Jun 01 2014
Journal Name
International Journal Of Physics And Research (ijpr)
CALCULATIONS OF MOLTIPOLE MIXING RATIOS FOR GAMMA TRANSITIONS OF Yb POPULATED FROM Yb REACTION USING -RATIO, CONSTANT STATISTICAL TENSOR AND LEAST SQUARES FITTING METHODS
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The - mixing ratios of -transitions from levels in populated in the reactions are calculated in present work using - ratio, constant statisticalTensor and least squares fitting methods The results obtained are in general, in good agreement or consistent, within the associated uncertainties, with these reported in Ref.[9],the discrepancies that occurs are due to inaccuracy existing in the experimental data The results obtained in the present work confirm the –method for mixed transitions better than that for pure transition because this method depends only on the experimental data where the second method depends on the pure or those considered to be pure -transitions, the same results occur in – method

Publication Date
Mon May 25 2026
Journal Name
Journal Mustansiriyah Of Sports Science
The effect of using Daniel's model for people with two types of brain control (left and right) to learn the skill of the Cartwheel in artistic gymnastics for second-stage students
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The research problem focused through the researcher's experience in the gymnastics game and the lack of use of educational models that give the student an important role in the educational process, so it became necessary to identify the type of prevailing style for students, and the need for diversity in the use of educational models based on scientific theories, including the Daniel Document model. Based on three theories of learning, which are structural, behavioral, and meaningful learning. The research aimed to identify the effect of using the Daniel model for people with two types of brain control (left and right) to learn the skill of the Cartwheel in artistic gymnastics for students of the second stage. The researcher used the experi

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Publication Date
Wed Mar 16 2022
Journal Name
Journal Of Educational And Psychological Researches
The Effectiveness of an Educational Program in Light Of Behavioral Cognitive Theory to Develop Efficient Response to Students Affected by Crises
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The aim of this research is to construct an educational program in light of the theory of behavioral cognitive and its impact on the development of the efficient response to students affected by crises (centers of your right to education). To achieve the objectives of the research, two scales were developed by the researcher in addition to two equivalent hypotheses were formulated. The scale contains (26) items divided into five fields; for its validity and reliability were derived based on the measure of efficient response, an educational program based on the theory of behavioral cognition. The test and the educational program were applied to a sample of (60) students from the centers of your right to education, divided into experimenta

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Publication Date
Thu Apr 01 2021
Journal Name
Basra Journal Of Science
Organic Field Effect Transistor Based on P3HT with Two Different Gate Dielectrics
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The electrical performance of bottom-gate/top source-drain contact for p-channel organic field-effect transistors (OFETs) using poly(3-hexylthiophene) (P3HT) as an active semiconductor layer with two different gate dielectric materials, Polyvinylpyrrolidone (PVP) and Hafnium oxide (HfO2), is investigated in this work. The output and transfer characteristics were studied for HfO2, PVP and HfO2/PVP as organic gate insulator layer. Both characteristics show a high drain current at the gate dielectric HfO2/PVP equal to -0.0031A and -0.0015A for output and transfer characteristics respectively, this can be attributed to the increasing of the dielectric capacitance. Transcondactance characteristics also studied for the three organic mater

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Publication Date
Sat Feb 02 2019
Journal Name
Journal Of The College Of Education For Women
Design of an Electron Gun Using two Electrod Low Aberration Immersion Lens
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Abstract
A two electrode immersion electrostatic lens used in the design
of an electron gun, with small aberration, has been designed using
the finite element method (FEM). By choosing the appropriate
geometrical shape of there electrodes the potential V(r,z) and the
axial potential distribution have been computed using the FEM to
solve Laplace's equation.
The trajectory of the electron beam and the optical properties of
this lens combination of electrodes have been computed under
different magnification conditions (Zero and infinite magnification
conditions) from studying the properties of the designed electron
gun can be supplied with Abeam current of 5.7*10-6 A , electron
gun with half acceptance

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